The bloom of economics and technology has had an enormous impact on society. Along with these developments, human activities nowadays produce massive amounts of data that can be easily collected for relatively low cost with the aid of new technologies. Many examples can be mentioned here including data from web term-document data, sensor arrays, gene expression, finance data, imaging and hyperspectral analysis. Because of the enormous amount of data from various different and new sources, more and more challenging scientific problems appear. These problems have changed the types of problems which mathematical scientists work. / In traditional statistics, the dimension of the data, p say, is low, with many observations, n say. In this case, classical rules such as the Central Limit Theorem are often applied to obtain some understanding from data. A new challenge to statisticians today is dealing with a different setting, when the data dimension is very large and the number of observations is small. The mathematical assumption now could be p > n, or even p goes to infinity and n fixed in many cases, for example, there are few patients with many genes. In these cases, classical methods fail to produce a good understanding of the nature of the problem. Hence, new methods need to be found to solve these problems. Mathematical explanations are also needed to generalize these cases. / The research preferred in this thesis includes two problems: Variable selection and Classification, in the case where the dimension is very large. The work on variable selection problems, in particular the Adaptive Lasso was completed by June 2007 and the research on classification has been carried out through out 2008 and 2009. The research on the Dantzig selector and the Lasso were finished in July 2009. Therefore, this thesis is divided into two parts. In the first part of the thesis we study the Adaptive Lasso, the Lasso and the Dantzig selector. In particular, in Chapter 2 we present some results for the Adaptive Lasso. Chapter 3 will provides two examples that show that neither the Dantzig selector or the Lasso is definitely better than the other. The second part of the thesis is organized as follows. In Chapter 5, we shall construct the model setting. In Chapter 6, we summarize the results of the scaled centroid-based classifier. We also prove some results on the scaled centroid-based classifier. Because there are similarities between the Support Vector Machine (SVM) and Distance Weighted Discrimination (DWD) classifiers, Chapter 8 introduces a class of distance-based classifiers that could be considered a generalization of the SVM and DWD classifiers. Chapters 9 and 10 are about the SVM and DWD classifiers. Chapter 11 demonstrates the performance of these classifiers on simulated data sets and some cancer data sets.

The bloom of economics and technology has had an enormous impact on society. Along with these developments, human activities nowadays produce massive amounts of data that can be easily collected for relatively low cost with the aid of new technologies. Many examples can be mentioned here including data from web term-document data, sensor arrays, gene expression, finance data, imaging and hyperspectral analysis. Because of the enormous amount of data from various different and new sources, more and more challenging scientific problems appear. These problems have changed the types of problems which mathematical scientists work. / In traditional statistics, the dimension of the data, p say, is low, with many observations, n say. In this case, classical rules such as the Central Limit Theorem are often applied to obtain some understanding from data. A new challenge to statisticians today is dealing with a different setting, when the data dimension is very large and the number of observations is small. The mathematical assumption now could be p > n, or even p goes to infinity and n fixed in many cases, for example, there are few patients with many genes. In these cases, classical methods fail to produce a good understanding of the nature of the problem. Hence, new methods need to be found to solve these problems. Mathematical explanations are also needed to generalize these cases. / The research preferred in this thesis includes two problems: Variable selection and Classification, in the case where the dimension is very large. The work on variable selection problems, in particular the Adaptive Lasso was completed by June 2007 and the research on classification has been carried out through out 2008 and 2009. The research on the Dantzig selector and the Lasso were finished in July 2009. Therefore, this thesis is divided into two parts. In the first part of the thesis we study the Adaptive Lasso, the Lasso and the Dantzig selector. In particular, in Chapter 2 we present some results for the Adaptive Lasso. Chapter 3 will provides two examples that show that neither the Dantzig selector or the Lasso is definitely better than the other. The second part of the thesis is organized as follows. In Chapter 5, we shall construct the model setting. In Chapter 6, we summarize the results of the scaled centroid-based classifier. We also prove some results on the scaled centroid-based classifier. Because there are similarities between the Support Vector Machine (SVM) and Distance Weighted Discrimination (DWD) classifiers, Chapter 8 introduces a class of distance-based classifiers that could be considered a generalization of the SVM and DWD classifiers. Chapters 9 and 10 are about the SVM and DWD classifiers. Chapter 11 demonstrates the performance of these classifiers on simulated data sets and some cancer data sets.

The goal of this project is to investigate any correlation between marketing emails and their receivers using machine learning and only a limited amount of initial data. The data consists of roughly 1200 emails and 98.000 receivers of these. Initially, the emails are grouped together based on their content using text clustering. They contain no information regarding prior labeling or categorization which creates a need for an unsupervised learning approach using solely the raw text based content as data. The project investigates state-of-the-art concepts like bag-of-words for calculating term importance and the gap statistic for determining an optimal number of clusters. The data is vectorized using term frequency - inverse document frequency to determine the importance of terms relative to the document and to all documents combined. An inherit problem of this approach is high dimensionality which is reduced using latent semantic analysis in conjunction with singular value decomposition. Once the resulting clusters have been obtained, the most frequently occurring terms for each cluster are analyzed and compared. Due to the absence of initial labeling an alternative approach is required to evaluate the clusters validity. To do this, the receivers of all emails in each cluster who actively opened an email is collected and investigated. Each receiver have different attributes regarding their purpose of using the service and some personal information. Once gathered and analyzed, conclusions could be drawn that it is possible to find distinguishable connections between the resulting email clusters and their receivers but to a limited extent. The receivers from the same cluster did show similar attributes as each other which were distinguishable from the receivers of other clusters. Hence, the resulting email clusters and their receivers are specific enough to distinguish themselves from each other but too general to handle more detailed information. With more data, this could become a useful tool for determining which users of a service should receive a particular email to increase the conversion rate and thereby reach out to more relevant people based on previous trends. / Målet med detta projekt att undersöka eventuella samband mellan marknadsföringsemail och dess mottagare med hjälp av oövervakad maskininlärning på en brgränsad mängd data. Datan består av ca 1200 email meddelanden med 98.000 mottagare. Initialt så gruperas alla meddelanden baserat på innehåll via text klustering. Meddelandena innehåller ingen information angående tidigare gruppering eller kategorisering vilket skapar ett behov för ett oövervakat tillvägagångssätt för inlärning där enbart det råa textbaserade meddelandet används som indata. Projektet undersöker moderna tekniker så som bag-of-words för att avgöra termers relevans och the gap statistic för att finna ett optimalt antal kluster. Datan vektoriseras med hjälp av term frequency - inverse document frequency för att avgöra relevansen av termer relativt dokumentet samt alla dokument kombinerat. Ett fundamentalt problem som uppstår via detta tillvägagångssätt är hög dimensionalitet, vilket reduceras med latent semantic analysis tillsammans med singular value decomposition. Då alla kluster har erhållits så analyseras de mest förekommande termerna i vardera kluster och jämförs. Eftersom en initial kategorisering av meddelandena saknas så krävs ett alternativt tillvägagångssätt för evaluering av klustrens validitet. För att göra detta så hämtas och analyseras alla mottagare för vardera kluster som öppnat något av dess meddelanden. Mottagarna har olika attribut angående deras syfte med att använda produkten samt personlig information. När de har hämtats och undersökts kan slutsatser dras kring hurvida samband kan hittas. Det finns ett klart samband mellan vardera kluster och dess mottagare, men till viss utsträckning. Mottagarna från samma kluster visade likartade attribut som var urskiljbara gentemot mottagare från andra kluster. Därav kan det sägas att de resulterande klustren samt dess mottagare är specifika nog att urskilja sig från varandra men för generella för att kunna handera mer detaljerad information. Med mer data kan detta bli ett användbart verktyg för att bestämma mottagare av specifika emailutskick för att på sikt kunna öka öppningsfrekvensen och därmed nå ut till mer relevanta mottagare baserat på tidigare resultat.